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The hardy types do not require any special attention over winter. It does help to leave the foilage as added protection during winter freezes, or you can cover with mulch. Once new growth begins in spring, any old growth may be pruned back.

Are they yellowing too? If so, your plants may be infected with fusarium wilt disease, which is common in asters and many other plants. Here is more information: https://www.gardeningknowhow.com/plant-problems/disease/controlling-fusarium-wilt.htm

From all the research I looked at, no it does not stay in the soil. It has to live in the host plant.
I think I would wait and make sure all the roots were gone from the location.
Here are a few articles.

https://www.gardeningknowhow.com/plant-problems/disease/aster-yellows-disease.htm
https://www.ag.ndsu.edu/cpr/plant-pathology/understanding-aster-yellows-7-26-12

Are they yellowing too? If so, your plants may be infected with fusarium wilt disease, which is common in asters and many other plants. Here is more information: https://www.gardeningknowhow.com/plant-problems/disease/controlling-fusarium-wilt.htm

A photo would aid in my identification as there is a wide range of worm-like pests that attack asters including beet armyworms, cabbage loopers, carnation leafrollers, corn earworms, cutworms and omnivorous leaftiers. Hand picking the pests as you have done is typically recommended although some might advocate the use of Bacillus thuringiensis.

Aster's do generally bloom later in the summer but you may have an issue with to much Nitrogen in the soil.
This will cause lush green plants with little to no flowering.

By adding Phosphorous you can promote flowering. Bone Meal is a good way to do this.

Here are some links for you.

https://www.gardeningknowhow.com/ornamental/flowers/asters/growing-asters.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/bone-meal-fertilizer.htm